Heterodyne weak measurements of nanorad beam deflections

Generally a measurement yields (eigen)values that can be distinguished and recorded by a suitable measuring device. Aharanov, Albert, and Vaidman (AAV) considered the case of weak measurements in quantum mechanics, where the eigenvaluespectrum is not properly resolved. AAV showed that in this case it is helpful to choose a suitable postselection state, which should be almost orthogonal to the initial state, and which is able to give rise to arbitrarily large expectation values. Here, we discuss how weak value amplification can be adapted to measure the optical activity of chiral liquids. The hallmark of optical activity is that the liquid has different refractive indices for right and left circularly polarized light, which causes the rotation of the plane of polarization of a linearly polarized light beam traversing the solution. This small difference (<< 10-7) in refractive indices may, not only be detected in transmission (polarimetry), but can also be detected in refraction. A linearly polarized light beam incident at an interface between a chiral and an achiral medium will split into its two circular polarization components, as the two components refract with different angles of refraction. A position sensitive detector may be used to register the difference in beam positions. For chemical and pharmaceutical applications it is of interest to measure small optical activities (with correspondingly small refractive index differences) in minute liquid samples. In order to increase the sensitivity of the measurement it becomes necessary to increase the separation between the two split circular beam components. We show that conventional weak value amplification, as reported in the literature before, does not provide sufficient information to determine the handedness of a chiral liquid. Instead we describe how weak value amplification combined with a heterodyne detection scheme can measure both the concentration and the handedness of the solution. The Faraday Effect in a glass prism is used as a model system for optical activity. The linear polarized beam is split into its circular polarization components at the interface.. To amplify these small angular displacements (typically a few nanoradians) of the two circular beam components we have implemented a weak measurement with a suitable post-selection scheme. It is shown, that the signal is proportional to the circular birefringence and furthermore sensitive to the sign of the optical activity.